This invention relates to imaging. More specifically, this invention relates to three dimensional computerized tomography providing for fast conversion of cone beam data to radon data.
In conventional computerized tomography for both medical and industrial applications, an x-ray fan beam and a linear array detector are employed. Two-dimensional (2D) imaging is achieved. While the data set is complete and image quality is correspondingly high, only a single slice of an object is imaged at a time. When a three-dimensional (3D) image is required, a "stack of slices" approach is employed. Acquiring a 3D data set a 2D slice at a time is inherently tedious and time-consuming. Moreover, in medical applications, motion artifacts occur because adjacent slices are not imaged simultaneously. Also, dose utilization is less than optimal because the distance between slices is typically less than the x-ray collimator aperture, resulting in double exposure to many parts of the body.
A more recent approach, based on what it called cone beam geometry, employs a two-dimensional array detector (often called an area detector) instead of a linear array detector, and a cone beam x-ray source instead of a fan beam x-ray source. At any instant, the entire object is irradiated by a cone beam x-ray source, and therefore cone beam scanning is much faster than slice/by/slice scanning using a fan beam or a parallel beam. Also, since each point in the object is viewed by the x-rays in 3D rather than in 2D, much higher contrast can be achieved than is possible with conventional 2D x-ray computerized tomography (CT). To acquire cone beam projection data, an object is scanned, preferably over a 360.degree. angular range. The scanning may be accomplished by moving the x-ray source in an appropriate scanning trajectory such as a circular trajectory around the object, while keeping the 2D array detector fixed with reference to the source. Alternately, the object may be rotated while the source and detector remain stationary. In either case, it is relative movement between the source and object which effects scanning.
Most image reconstruction procedures in x-ray CT are based on the Radon inversion process, in which the image of an object is reconstructed from the totality of the Radon transform of the object. The radon transform of a 3D object consists of planar integrals. The cone beam data, however, are not directly compatible with image reconstruction through inverse radon transformation, which requires the use of planar integrals of the object as input. Consequently, image reconstruction by inversion from cone beam scanning data generally comprises two steps. A first step is to convert the cone beam data to planar integrals. A second step is then to perform an inverse Radon transform on the planar integrals to obtain the image.
The first step is described by the present inventor's prior application Ser. No. 07/631,815, filed Dec. 21, 1990, entitled "METHOD AND APPARATUS FOR CONVERTING CONE BEAM X-RAY PROJECTION DATA TO PLANAR INTEGRALS AND RECONSTRUCTING A THREE-DIMENSIONAL COMPUTERIZED TOMOGRAPHY (CT) IMAGE OF AN OBJECT", assigned to the assignee of the present application and hereby incorporated by reference. A technique for performing an inverse Radon transform on planar integrals to obtain an image is described in the present inventor's prior U.S. patent application Ser. No. 07/631,818, filed Dec. 21, 1990, entitled "PARALLEL PROCESSING METHOD AND APPARATUS FOR RECONSTRUCTING A THREE-DIMENSIONAL COMPUTERIZED TOMOGRAPHY (CT) IMAGE OF AN OBJECT FROM CONE BEAM PROJECTION DATA OR FROM PLANAR INTEGRALS", assigned to the assignee of the present application, and hereby incorporated by reference. Thus, those two prior incorporated by reference U.S. patent applications describe techniques which may be used for three-dimensional image reconstruction by inversion from cone beam scanning data.
Further image reconstruction techniques are disclosed in the present inventor's prior U. S. patent application Ser. No. 07/631,514, filed Dec. 21, 1990, entitled "METHOD AND APPARATUS FOR RECONSTRUCTING A THREE-DIMENSIONAL COMPUTERIZED TOMOGRAPHY (CT) IMAGE OF AN OBJECT FROM INCOMPLETE CONE BEAM PROJECTION DATA", assigned to the assignee of the present application, and hereby incorporated by reference.
One method for converting cone beam data to planar integrals is disclosed in Gerald N. Minerbo, "Convolutional Reconstruction from Cone-Beam Projection Data", IEEE Trans. Nucl. Sci. , Vol. NS-26, No. 2, pp. 2682-2684 (April 1979). Unfortunately, as is discussed, for example, in L. A. Fledkamp, L. C. Davis, and J. W. Kress, "Practical Cone-Beam Algorithm", J. Opt. Soc. Am. A. , Vol. 1, No. 6, pp. 612-619 (June 1984) , the derivation in Minerbo contains an error which cannot easily be rectified and which renders the result invalid.
In Bruce D. Smith, "Image Reconstruction from Cone-Beam Projections: Necessary and Sufficient Conditions and Reconstruction Methods", IEEE Trans. Med. Imag., Vol. MI-44, pp. 1425 (March 1985), there is disclosed a method for converting from cone beam data the one-dimensional convolution of the planar integrals with the Horn's kernel. Since the convolution mixes together the planar integrals on all the planes, the computation of one point of the convolved result requires all the data on the detector at one view angle. Thus the task is very computationally intensive.
In P. Grangeat, "Analysis of A 3D Imaging System by Reconstruction from X Radiographies in Conical Geometry" ("Analyse d'un System D-Imagerie 3D par Reconstruction a par-tir de Radiographies X en Geometrie conique"), Ph.D. Thesis, National College of Telecommunications (I-Ecole Nationale Superieur des Telecommunications), France (1987), a technique is disclosed for computing the derivative of the planar integrals from cone beam data. The computed data points, however, reside on a set of great circles on a spherical shell in Radon space. These great circles in general do not fall on any arbitrary set of planes in Radon spaces, and do not fall on a set of coaxial vertical planes in Radon space. Thus they are not suitable for input to inverse Radon transformation. It would require an extensive effort in three-dimensional interpolation to get the data on the vertical planes to be used in inverse Radon transformation, and furthermore interpolation would introduce errors into the data.
Although the present inventor's prior U.S. application Ser. No. 07/631,815 describes a technique for converting the cone beam data to planar integrals (so that an inverse Radon transform may be performed on the planar integrals to obtain the image), the technique requires approximately N.sup.4 operations to be performed where N.sup.3 is the number of Radon points. Specifically, a line integral is calculated for each of the Radon points and the computation of that line integral for each Radon point involves approximately N operations. Although this technique is generally satisfactory, the large number of operations are time consuming. A relatively large amount of processing capacity may be required in order to perform the large number of operations sufficiently quickly to produce real time three-dimensional images.